# Properties

 Label 3332.1.o.f Level $3332$ Weight $1$ Character orbit 3332.o Analytic conductor $1.663$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -68 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.66288462209$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 476) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.188737808.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{2} q^{2} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{3} + \zeta_{12}^{4} q^{4} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{6} - q^{8} + ( - \zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{9}+O(q^{10})$$ q + z^2 * q^2 + (-z^5 - z^3) * q^3 + z^4 * q^4 + (-z^5 + z) * q^6 - q^8 + (-z^4 - z^2 - 1) * q^9 $$q + \zeta_{12}^{2} q^{2} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{3} + \zeta_{12}^{4} q^{4} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{6} - q^{8} + ( - \zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{9} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{11} + (\zeta_{12}^{3} + \zeta_{12}) q^{12} - q^{13} - \zeta_{12}^{2} q^{16} - \zeta_{12}^{4} q^{17} + ( - \zeta_{12}^{4} - \zeta_{12}^{2} + 1) q^{18} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{22} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{24} + \zeta_{12}^{4} q^{25} - \zeta_{12}^{2} q^{26} + (\zeta_{12}^{5} + \zeta_{12}) q^{27} - \zeta_{12}^{4} q^{32} + ( - \zeta_{12}^{4} - 2 \zeta_{12}^{2} - 1) q^{33} + q^{34} + ( - \zeta_{12}^{4} + \zeta_{12}^{2} + 1) q^{36} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{39} + (\zeta_{12}^{3} + \zeta_{12}) q^{44} + (\zeta_{12}^{5} - \zeta_{12}) q^{48} - q^{50} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{51} - \zeta_{12}^{4} q^{52} - \zeta_{12}^{4} q^{53} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{54} + q^{64} + ( - \zeta_{12}^{4} - \zeta_{12}^{2} + 1) q^{66} + \zeta_{12}^{2} q^{68} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{71} + (\zeta_{12}^{4} + \zeta_{12}^{2} + 1) q^{72} + (\zeta_{12}^{3} + \zeta_{12}) q^{75} + (\zeta_{12}^{5} - \zeta_{12}) q^{78} + (\zeta_{12}^{3} + \zeta_{12}) q^{79} + (\zeta_{12}^{4} + 1) q^{81} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{88} + \zeta_{12}^{2} q^{89} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{96} + (2 \zeta_{12}^{5} - \zeta_{12}^{3} - 2 \zeta_{12}) q^{99} +O(q^{100})$$ q + z^2 * q^2 + (-z^5 - z^3) * q^3 + z^4 * q^4 + (-z^5 + z) * q^6 - q^8 + (-z^4 - z^2 - 1) * q^9 + (-z^5 - z^3) * q^11 + (z^3 + z) * q^12 - q^13 - z^2 * q^16 - z^4 * q^17 + (-z^4 - z^2 + 1) * q^18 + (-z^5 + z) * q^22 + (z^5 + z^3) * q^24 + z^4 * q^25 - z^2 * q^26 + (z^5 + z) * q^27 - z^4 * q^32 + (-z^4 - 2*z^2 - 1) * q^33 + q^34 + (-z^4 + z^2 + 1) * q^36 + (z^5 + z^3) * q^39 + (z^3 + z) * q^44 + (z^5 - z) * q^48 - q^50 + (-z^3 - z) * q^51 - z^4 * q^52 - z^4 * q^53 + (-z^3 - z) * q^54 + q^64 + (-z^4 - z^2 + 1) * q^66 + z^2 * q^68 + (-z^5 + z) * q^71 + (z^4 + z^2 + 1) * q^72 + (z^3 + z) * q^75 + (z^5 - z) * q^78 + (z^3 + z) * q^79 + (z^4 + 1) * q^81 + (z^5 + z^3) * q^88 + z^2 * q^89 + (-z^3 - z) * q^96 + (2*z^5 - z^3 - 2*z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 4 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^4 - 4 * q^8 - 4 * q^9 $$4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 4 q^{9} - 4 q^{13} - 2 q^{16} + 2 q^{17} + 4 q^{18} - 2 q^{25} - 2 q^{26} + 2 q^{32} - 6 q^{33} + 4 q^{34} + 8 q^{36} - 4 q^{50} + 2 q^{52} + 2 q^{53} + 4 q^{64} + 6 q^{66} + 2 q^{68} + 4 q^{72} - 2 q^{81} + 2 q^{89}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^4 - 4 * q^8 - 4 * q^9 - 4 * q^13 - 2 * q^16 + 2 * q^17 + 4 * q^18 - 2 * q^25 - 2 * q^26 + 2 * q^32 - 6 * q^33 + 4 * q^34 + 8 * q^36 - 4 * q^50 + 2 * q^52 + 2 * q^53 + 4 * q^64 + 6 * q^66 + 2 * q^68 + 4 * q^72 - 2 * q^81 + 2 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0.500000 + 0.866025i −0.866025 + 1.50000i −0.500000 + 0.866025i 0 −1.73205 0 −1.00000 −1.00000 1.73205i 0
67.2 0.500000 + 0.866025i 0.866025 1.50000i −0.500000 + 0.866025i 0 1.73205 0 −1.00000 −1.00000 1.73205i 0
2039.1 0.500000 0.866025i −0.866025 1.50000i −0.500000 0.866025i 0 −1.73205 0 −1.00000 −1.00000 + 1.73205i 0
2039.2 0.500000 0.866025i 0.866025 + 1.50000i −0.500000 0.866025i 0 1.73205 0 −1.00000 −1.00000 + 1.73205i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by $$\Q(\sqrt{-17})$$
4.b odd 2 1 inner
7.c even 3 1 inner
17.b even 2 1 inner
28.g odd 6 1 inner
119.j even 6 1 inner
476.o odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.o.f 4
4.b odd 2 1 inner 3332.1.o.f 4
7.b odd 2 1 476.1.o.c 4
7.c even 3 1 3332.1.g.f 2
7.c even 3 1 inner 3332.1.o.f 4
7.d odd 6 1 476.1.o.c 4
7.d odd 6 1 3332.1.g.g 2
17.b even 2 1 inner 3332.1.o.f 4
28.d even 2 1 476.1.o.c 4
28.f even 6 1 476.1.o.c 4
28.f even 6 1 3332.1.g.g 2
28.g odd 6 1 3332.1.g.f 2
28.g odd 6 1 inner 3332.1.o.f 4
68.d odd 2 1 CM 3332.1.o.f 4
119.d odd 2 1 476.1.o.c 4
119.h odd 6 1 476.1.o.c 4
119.h odd 6 1 3332.1.g.g 2
119.j even 6 1 3332.1.g.f 2
119.j even 6 1 inner 3332.1.o.f 4
476.e even 2 1 476.1.o.c 4
476.o odd 6 1 3332.1.g.f 2
476.o odd 6 1 inner 3332.1.o.f 4
476.q even 6 1 476.1.o.c 4
476.q even 6 1 3332.1.g.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.1.o.c 4 7.b odd 2 1
476.1.o.c 4 7.d odd 6 1
476.1.o.c 4 28.d even 2 1
476.1.o.c 4 28.f even 6 1
476.1.o.c 4 119.d odd 2 1
476.1.o.c 4 119.h odd 6 1
476.1.o.c 4 476.e even 2 1
476.1.o.c 4 476.q even 6 1
3332.1.g.f 2 7.c even 3 1
3332.1.g.f 2 28.g odd 6 1
3332.1.g.f 2 119.j even 6 1
3332.1.g.f 2 476.o odd 6 1
3332.1.g.g 2 7.d odd 6 1
3332.1.g.g 2 28.f even 6 1
3332.1.g.g 2 119.h odd 6 1
3332.1.g.g 2 476.q even 6 1
3332.1.o.f 4 1.a even 1 1 trivial
3332.1.o.f 4 4.b odd 2 1 inner
3332.1.o.f 4 7.c even 3 1 inner
3332.1.o.f 4 17.b even 2 1 inner
3332.1.o.f 4 28.g odd 6 1 inner
3332.1.o.f 4 68.d odd 2 1 CM
3332.1.o.f 4 119.j even 6 1 inner
3332.1.o.f 4 476.o odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$:

 $$T_{3}^{4} + 3T_{3}^{2} + 9$$ T3^4 + 3*T3^2 + 9 $$T_{5}$$ T5 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 3T^{2} + 9$$
$13$ $$(T + 1)^{4}$$
$17$ $$(T^{2} - T + 1)^{2}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} - T + 1)^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$(T^{2} - 3)^{2}$$
$73$ $$T^{4}$$
$79$ $$T^{4} + 3T^{2} + 9$$
$83$ $$T^{4}$$
$89$ $$(T^{2} - T + 1)^{2}$$
$97$ $$T^{4}$$